Forbidden pairs for -connected Hamiltonian graphs

For an integer k with k ≥ 2 and a pair of connected graphs F 1 and F 2 of order at least three, we say that { F 1 , F 2 } is a k -forbidden pair if every k -connected { F 1 , F 2 } -free graph, except possibly for a finite number of exceptions, is Hamiltonian. If no exception arises, { F 1 , F 2 } is said to be a strong k -forbidden pair. The 2-forbidden pairs and the strong 2-forbidden pairs are determined by Faudree and Gould (1997) [11] and Bedrossian (1991) [1], respectively. All of them contain K 1 , 3 . In this paper, we prove that { K 1 , k + 1 , P 4 } is a strong k -forbidden pair, which shows that K 1 , 3 is not always necessary in a k -forbidden pair for k ≥ 3 . On the other hand, we prove that each k -forbidden pair contains K 1 , l for some l ≤ k + 1 . We also discuss several other Hamiltonian properties of k -connected { K 1 , k + 1 , P 4 } -free graphs.